Quantum Causal Graph Dynamics

Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded speed, with respect to the distance given by the graph. Suppose, moreover, that the graph itself is subject to the evolution, and may be driven to be in a quantum superposition of graphs---in accordance to the superposition principle. We show that these unitary causal operators must decompose as a finite-depth circuit of local unitary gates. This unifies a result on Quantum Cellular Automata with another on Reversible Causal Graph Dynamics. Along the way we formalize a notion of causality which is valid in the context of quantum superpositions of time-varying graphs, and has a number of good properties. Keywords: Quantum Lattice Gas Automata, Block-representation, Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum Graphity, Causal Dynamical Triangulations, Spin Networks, Dynamical networks, Graph Rewriting.Comment: 8 pages, 1 figur

Causal graph dynamics

We extend the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time - but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalized. The definition we provide for these "causal graph dynamics" is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of Cellular Automata theory carry through: causal graph dynamics admit a characterization as continuous functions, and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics. KEYWORDS: Dynamical networks, Boolean networks, Generative networks automata, Cayley cellular automata, Graph Automata, Graph rewriting automata, Parallel graph transformations, Amalgamated graph transformations, Time-varying graphs, Regge calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3: Typos corrected, figure adde

Index theory of one dimensional quantum walks and cellular automata

If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the sense that there is a system S which locally acts like S_1 in one region and like S_2 in some other region, if and only if S_1 and S_2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S_1 into S_2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S -> ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.Comment: 38 pages. v2: added examples, terminology clarifie

Energy-based Analysis of Biochemical Cycles using Bond Graphs

Thermodynamic aspects of chemical reactions have a long history in the Physical Chemistry literature. In particular, biochemical cycles - the building-blocks of biochemical systems - require a source of energy to function. However, although fundamental, the role of chemical potential and Gibb's free energy in the analysis of biochemical systems is often overlooked leading to models which are physically impossible. The bond graph approach was developed for modelling engineering systems where energy generation, storage and transmission are fundamental. The method focuses on how power flows between components and how energy is stored, transmitted or dissipated within components. Based on early ideas of network thermodynamics, we have applied this approach to biochemical systems to generate models which automatically obey the laws of thermodynamics. We illustrate the method with examples of biochemical cycles. We have found that thermodynamically compliant models of simple biochemical cycles can easily be developed using this approach. In particular, both stoichiometric information and simulation models can be developed directly from the bond graph. Furthermore, model reduction and approximation while retaining structural and thermodynamic properties is facilitated. Because the bond graph approach is also modular and scaleable, we believe that it provides a secure foundation for building thermodynamically compliant models of large biochemical networks

The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy

The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the elements in the distribution. Because classical entropy-based Maxent collapses cases confounding all distinct degrees of randomness and pseudo-randomness, here we take into consideration the generative mechanism of the systems considered in the ensemble to separate objects that may comply with the principle under some restriction and whose entropy is maximal but may be generated recursively from those that are actually algorithmically random offering a refinement to classical Maxent. We take advantage of a causal algorithmic calculus to derive a thermodynamic-like result based on how difficult it is to reprogram a computer code. Using the distinction between computable and algorithmic randomness we quantify the cost in information loss associated with reprogramming. To illustrate this we apply the algorithmic refinement to Maxent on graphs and introduce a Maximal Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a generalisation over previous approaches. We discuss practical implications of evaluation of network randomness. Our analysis provides insight in that the reprogrammability asymmetry appears to originate from a non-monotonic relationship to algorithmic probability. Our analysis motivates further analysis of the origin and consequences of the aforementioned asymmetries, reprogrammability, and computation.Comment: 30 page

Quantum Trajectories, State Diffusion and Time Asymmetric Eventum Mechanics

We show that the quantum stochastic unitary dynamics Langevin model for continuous in time measurements provides an exact formulation of the Heisenberg uncertainty error-disturbance principle. Moreover, as it was shown in the 80's, this Markov model induces all stochastic linear and non-linear equations of the phenomenological "quantum trajectories" such as quantum state diffusion and spontaneous localization by a simple quantum filtering method. Here we prove that the quantum Langevin equation is equivalent to a Dirac type boundary-value problem for the second-quantized input "offer waves from future" in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the "trajectories of the output particles". This result supports the wave-particle duality in the form of the thesis of Eventum Mechanics that everything in the future is constituted by quantized waves, everything in the past by trajectories of the recorded particles. We demonstrate how this time arrow can be derived from the principle of quantum causality for nondemolition continuous in time measurements.Comment: 21 pages. See also relevant publications at http://www.maths.nott.ac.uk/personal/vpb/publications.htm